With strong direct and indirect network effects and a large user base built on the back of the popular eBay platform PayPal of the late 2000s looked well-positioned to dominate online money transfers and become synonymous with online payments in the same way in which Google and Facebook control search and social networking. 
Unfortunately for PayPal, the emergence of cheap and convenient credit card processing as well as the shift towards mobile have since then undermined the strength of indirect and direct effects and have opened the door to the online payment market to smaller competitors as well as to the behemoths of the digital economy.
Indirect network effects weaken as alternatives emerge
In the early days of PayPal a consumer’s decision to join the platform was driven not just by eBay’s push for using PayPal for payment processing (PayPal used to be the only way to pay other users on eBay) but also by the growing number of retailers who accepted PayPal as their checkout method of choice. On the other side of the market, the large number of customers using PayPal made it a clear winner for processing online payments for retailers and aspiring retailers.
However, as retailers gained the ability to accept credit cards at little cost (often facilitated by new companies such as Square or Intuit as well as PayPal’s introduction in June 2010 of Guest Payment credit card processing services – more on this here) the number of parties on the other side of the platform ceased to matter as much. When deciding between platforms, businesses no longer need to care as much about how many customers were on each platform as they knew most customers could just pay with a credit card. Similarly, most customers no longer needed to pay much attention about which platform was more popular with retailers as they knew they would always have some way of paying.
Direct network effects weaken as new jobs emerge
When facilitating online payments between strangers selling items through eBay auctions, the overall size of the network played an important role and new entrants would find it difficult to compete with the strength of the direct network effects playing in PayPal’s favor. However, the growing use of smartphones brought about a new job to be done for online payments providers: the facilitation of money transfers between people from the same real-world networks. For this job, the penetration of one’s close network mattered far more than the overall size of the network which allowed smaller players benefitting from viral spread  and seamless mobile experiences such as Venmo (actually acquired by PayPal as part of its acquisition of Braintree in September 2013) or Popmoney to grow quickly within certain communities and take on much larger competitors. 
Multi-homing deters winner-take-all
The shift to greater use of mobile apps and the improvements in user experiences also helped consumers multi-home across different platforms which made peer-to-peer payments even more contestable and further challenged the tendency towards a winner-takes-all market structure fueled initially by the strong network effects.
Scale may matter but PayPal not alone
Although the network effects may have weakened in the online payments business, the fixed cost nature of the business makes PayPal’s scale an important advantage relative to some of its smaller competitors. This advantage may however not be there vis-à-vis the payment offering from Apple and Google that are integrating their own online (and increasingly offline) payment solutions with the devices powered by their operating systems. Although both Apple Pay and Google Wallet have initially stayed away from peer-to-peer payments this is clearly a natural extension of the services which may provide a major challenge to incumbents in the online payments space. In fact, Google repurposed Google Wallet exactly for this purpose following the introduction of Android Pay just a few weeks ago – more on this here.
While PayPal deserves credit for trying to stay on the front foot of the online payments business through its acquisitions, there’s no guarantee these will be sufficient to deliver the monopoly position PayPal may once have hoped for. With PayPal’s share price below its July 20th IPO price, investors are certainly yet to be convinced.
 PayPal’s cofounder Peter Thiel has much to say about online monopolies. See for example his Wall Street Journal contribution Competition is for losers.
 People who use Venmo will urge one to set it up on one’s phone by the time a shared taxi gets to its final destination.
 Investopedia has a good piece on Venmo’s business model and competition.